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Multiple elliptic integrals and differential equations

John M. Campbell, M. Lawrence Glasser, Yajun Zhou

Abstract

We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$.

Multiple elliptic integrals and differential equations

Abstract

We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral with integrals involving .
Paper Structure (3 sections, 1 theorem, 29 equations)

This paper contains 3 sections, 1 theorem, 29 equations.

Table of Contents

  1. Introduction
  2. Multiple elliptic integrals via a third-order ODE
  3. Multiple elliptic integrals via a second-order PDE

Key Result

Lemma 2.1

We have an integral identity

Theorems & Definitions (9)

  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['eq:intODE']} by ODE
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • proof : Proof of \ref{['eq:intPDE']} by PDE